Some Studies On (r,s) Entropy

(r,s)Entropy,as a generalization of multi-parameter entropy,its properties have important reference significance for the research of other generalized entropy.In this paper,we will continue to study(r,s)entropy on the basis of previous studies.Firstly,the tight upper bounds of(r,s)entropy for classical and quantum cases are established by using the probabilistic coupling method,the stability of entropy is studied.Secondly,in the class of life distribution,the cumulative residual(r,s)entropy based on survival function,dynamic cumulative residual(r,s)entropy,weighted cumulative residual(r,s)entropy and dynamic weighted cumulative residual(r,s)entropy are defined respectively to measure the uncertainty of residual life,and the weighted version emphasizes the importance of the larger of the random variables.In this paper,some properties of these entropies and their relations with each other,as well as with the risk rate function and the average residual life function are studied.The cumulative residual entropy with multi-parameters has reference significance to the cumulative residual entropy based on Renyi entropy,Tsallis entropy and other single-parameter.Finally,we define and study the weighted cumulative residual(r,s)entropy of the order statistics,and its related properties.First,we prove the relationship between the weighted cumulative residual(r,s)entropy of order statistics and the size of statistics,risk function,record value,etc.Then we show the change of weighted cumulative residual(r,s)entropy of order statistics in linear transformation,the monotonicity of the entropy is also discussed.